Download presentation

Presentation is loading. Please wait.

Published byOmari Pittmon Modified over 2 years ago

1
Chapter 13-Vector Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

3
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

4
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

5
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let v denote the velocity of a wind that blows through a wind tunnel parallel to the sides of the tunnel. Suppose that at each point P = (x, y, z), the magnitude of v (P) is proportional to the height of P above the ground. Assume that the ground is at height z = 0 and that the vector i points down the tunnel in the direction that the wind is blowing. Describe the velocity as a vector field. Vector Fields in Physics

6
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Determine the field lines of the vector field F(x, y) = i + y j that is shown below. Integral Curves (Streamlines)

7
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let be a positive constant. The differential equation is called the harmonic oscillator equation. It is known that u (t) is a solution of this equation if and only if u (t) = Acos ( t) + B sin ( t) for some constants A and B. Use this fact to determine the integral curves of the vector field F(x, y) = − yi + xj. Integral Curves (Streamlines)

8
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let the temperature at a point in the plane be given by T(x, y) = x 2 +6y 2. Calculate the gradient of T and give a physical interpretation. Gradient Vector Fields and Potential Functions

9
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Gradient Vector Fields and Potential Functions

10
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Gradient Vector Fields and Potential Functions

11
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: The vector field F(x, y)=(y − 3x 2 ) i + (x + sin (y)) j is known to be conservative. Find a scalar valued function u for which F = u. Finding Potential Functions EXAMPLE: The vector field F(x, y, z) = y 2 i + (2xy + z) j + (y + 3) k is known to be conservative. Find a scalar valued function u for which F = u.

12
Chapter 13-Vector Calculus 13.1 Vector Fields Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. Is {(x, y, z) : 0 < x, 0 < y, 0 < z} a region? If the answer is No, then why not? 2. Is {(x, y, z) : 0 < xyz} a region? If the answer is No, then why not? 3. True or false: If a particle is in motion due to a conservative force field, then the kinetic energy of the particle is conserved. 4. The vector field F(x, y) = y i+(x + 3) j is known to be conservative. Find a scalar-valued function u for which F = u. Quick Quiz

13
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals DEFINITION: Let C be a directed curve (in the plane or in space) with parameterization t r(t), a≤ t ≤ b. Suppose that F is a continuous vector field defined on C. Then the line integral (or path integral) of F over C is denoted by the symbol C F· dr and defined to be

14
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals EXAMPLE: Let C be the planar directed curve parameterized by r(t) =, 0 ≤ t ≤ 1. Calculate R F· dr for F(x, y) = e x i + xyj.

15
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals THEOREM: Let t r(t), a ≤ t ≤ b be a continuously differentiable parameterization for a directed curve C in the plane or in space. Suppose that F is a continuous vector field whose domain contains C. Let s p(s), 0 ≤ s ≤ L be the arc length parameterization of C with p (0) = r (a) and p (L) = r (b). Let T(s) = p’ (s) denote the unit tangent vector to C at the point p(s). Then the line integral of F over C is given by the equation

16
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Line Integrals

17
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Dependence on Path EXAMPLE: Let F(x, y) = −y i + x j. Set P 0 = (1, 0) and P 1 = (−1, 0). Consider two paths from P 0 to P 1 : C parameterized by r(t) = cos (t) i + sin (t) j, 0 ≤ t ≤ and C * parameterized by r * (t) = cos (t) i − sin (t) j, 0 ≤ t ≤ . Is the line integral of F over C equal to the line integral of F over C * ?

18
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Curves EXAMPLE: Calculate where C comprises the line segment from P 0 = (1, 0, 0) to P 1 = (0, 1, 0), the line segment from P 1 to P 2 = (0, 0, 2), and the line segment from P 2 to P 0.

19
Chapter 13-Vector Calculus 13.2 Line Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

20
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

21
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

22
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

23
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Consider the vector field F(x, y, z) = yzi + xzj + xyk. Show that F is path independent. Calculate the line integral of F from P 0 = (0,−1, 2) to P 1 = (2, 1, 4).

24
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Characterization of Path-Independent Vector Fields

25
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Characterization of Path-Independent Vector Fields

26
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Vector Fields

27
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field F(x, y) = (x 2 − sin (y))i +(y 3 + cos (x))j conservative? Closed Vector Fields

28
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Closed Vector Fields

29
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let M (x, y) = sin (y) + y sin (x) and N (x, y) = x cos (y) − cos (x) + 1. Is the vector field F(x, y) = M (x, y) i + N (x, y) j conservative on the rectangle G = {(x, y) : −1 < x < 1,−1 < y < 1}? If it is, then find a function u such that F = u. Closed Vector Fields

30
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vector Fields in Space

31
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field in space given by F(x, y, z) = xz i + yz j +(y 2 /2) k a conservative vector field? Vector Fields in Space

32
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved THEOREM: Suppose that F(x, y, z) = M(x, y, z)i+ N(x, y, z)j + R(x, y, z) k is a continuously differentiable vector field on a simply connected region G in space. If F satisfies all three equations then there is a twice continuously differentiable function u on G such that u = F. In short, a closed vector field on a simply connected region in space is conservative. Vector Fields in Space

33
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Is the vector field F(x, y, z) = yz 2 i + (xz2−z)j + (2xyz−y) k conservative on the box G = {(x, y, z) : 0 < x < 2, 0 < y < 2, 0 < z < 2}? If it is, find a function u for which F = u. Vector Fields in Space

34
Chapter 13-Vector Calculus 13.3 Conservative Vector Fields and Path Independence Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

35
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence of a Vector Field

36
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence of a Vector Field EXAMPLE: Define F(x, y) = xi + yj, G(x, y) = −xi − y 3 j, and H(x, y) = x 2 i − y 2 j. Calculate the divergence of each vector field at the origin and relate your answer to the physical properties of the flow that the vector field represents.

37
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Curl of a Vector Field EXAMPLE: Define F(x, y) = xi + yj, G(x, y) = −xi − y 3 j, and H(x, y) = x 2 i − y 2 j. Calculate the divergence of each vector field at the origin and relate your answer to the physical properties of the flow that the vector field represents.

38
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Curl of a Vector Field EXAMPLE: Define F(x, y, z) = y 2 z i − x 3 j + xy k. Calculate curl(F). EXAMPLE: Sketch the vector field F(x, y, z) = −yi + xj + 0k and its curl.

39
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Identities Involving div, curl, grad, and THEOREM: Let u be a twice continuously differentiable scalar-valued function on a region G in the plane or in space. Let F be a twice continuously differentiable vector field on G. Then i) div( grad (u)) = u, ii) div(curl (F)) = 0, iii) curl(grad (u)) = 0, iv) curl(curl (F)) =grad(div (F)) − F, v) div(uF) = u div(F) +grad (u) · F.

40
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Identities Involving div, curl, grad, and EXAMPLE: For F(x, y, z) =, verify the identity curl(curl (F)) =grad(div (F)) − F.

41
Chapter 13-Vector Calculus 13.4 Divergence, Gradient, and Curl Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Calculate the divergence of x 2 yzi + 7yj − xyk at the point (1, 2,−1). 2. Calculate the curl of x 2 i + (12/z) j − y 3 k at the point (−4, 3, 2). 3. Calculate × ( u) for u (x, y, z) = x 2 y 3 + z 4. 4. True or false: F is conservative implies curl(F) = 0.

42
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

43
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

44
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

45
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the area A inside the ellipse x 2 /a 2 + y 2 /b 2 = 1.

46
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved A Vector Form of Green’s Theorem EXAMPLE: Let F(x, y) = x i + y j and R = {(x, y) : x 2 + y 2 < 1}. Assume that F represents the velocity of the flow of a fluid in the region. Explain what Green’s Theorem tells us about this fluid flow.

47
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Green’s Theorem for More General Regions

48
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Green’s Theorem for More General Regions

49
Chapter 13-Vector Calculus 13.5 Green’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. True or false: If a simple closed curve C is part of the boundary of a region, then the positive orientation of C is always counterclockwise. 2. If C is a positively oriented simple closed curve that encloses a region of area 3 and if F(x, y) = 5y i − 2x j, then what is the value of 3. If C is a positively oriented simple closed curve that encloses a region of area 3, if F(x, y) = 7x i − 5y j, and if n is the outward unit normal along C, then what is the value of

50
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral for Surface Area DEFINITION: Let S be the graph of a continuously differentiable function (x, y) f (x, y) defined on a region R in the xy-plane. We refer to as the element of surface area on S. The surface area of the graph of f over R is defined by

51
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral for Surface Area EXAMPLE: Find the area of the portion of the plane 3x + 2y + z = 6 that lies over the interior of the circle x 2 + y 2 = 1 in the xy-plane.

52
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integrating a Function over a Surface

53
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Integrating a Function over a Surface

54
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application EXAMPLE: Assuming that it has uniform mass distribution , determine the center of mass of the upper half of the sphere x 2 + y 2 + z 2 = a 2.

55
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Element of Area for a Surface that is Given Parametrically

56
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Integrals Over Parameterized Surfaces

57
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Surface Integrals Over Parameterized Surfaces EXAMPLE: Integrate the function (x, y, z) = z − xy over the surface S that is parameterized by r(u, v) =, 0 ≤u ≤ 1, 0 ≤ v ≤ 2.

58
Chapter 13-Vector Calculus 13.6 Surface Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

59
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Orientable Surfaces and Their Boundaries EXAMPLE: Suppose that a > 0. What are the two orientations of the sphere x 2 + y 2 + z 2 = a 2 ?

60
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Component of Curl in the Normal Direction

61
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stoke’s Theorem

62
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stoke’s Theorem

63
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stokes’s Theorem on a Region with Piecewise Smooth Boundary

64
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Stokes’s Theorem on a Region with Piecewise Smooth Boundary

65
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application

66
Chapter 13-Vector Calculus 13.7 Stoke’s Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

67
Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence Theorem

68
Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Divergence Theorem

69
Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Some Applications

70
Chapter 13-Vector Calculus 13.8 Divergence Theorem Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

Similar presentations

Presentation is loading. Please wait....

OK

Operators in scalar and vector fields

Operators in scalar and vector fields

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on operating system deadlock Ppt on op amp circuits equations Ppt on bluetooth hacking statistics Ppt on albert einstein Ppt on electricity for class 10th Ppt on world environment day theme Ppt on needle stick injury management Ppt on operational research Ppt on limits and continuity of functions Ppt on object-oriented programming interview questions